Collaborative Research Center SFB559 Modeling of Large Logistic Networks Project M8 - Optimization Rich Vehicle Routing Problems Challenges and Prospects in Exploring the Power of Parallelism Andreas Reinholz 1 st COLLAB Workshop 12.04.2010
Structure Motivation and context Metaheuristics as Iterative Variation Selection Procedures Elementary and composed Neighborhood Generating Operators Modeling concepts and constraint handling Neighborhood Generating Operators for Vehicle Routing Problems Acceleration techniques and efficient data structures Decomposition methods Closer to the real world: Modeling uncertainty, flexibility and risk Conclusions and outlook 2
Features and Challenges of Logistics Optimization Tasks Mixed Integer Optimization Problems with Various constraints Multiple objectives Range from Strategic Planning to Online-Optimization Open or Disturbed Systems, imprecise or incomplete data, noise Dynamic Optimization tasks with moving optima Hierarchies of complex optimization problems Integration in Interactive Decision Support Systems Evaluation model could be a Simulation Model or a Black Box 3
Metaheuristics Neighborhood Search (NS) Variable Neighborhood Search (VNS) Iterative Local Search (ILS) (Recursive) Iterative Local Search (R-ILS) Tabu Search (TS) Greedy Randomized Adaptive Search Procedure (GRASP) Evolutionary Algorithms (EA) Ant-Systems, Particle Swarm, Scatter Search Adaptive Memory Programming Estimation of Distribution Algorithms (EDA) Multiple Agent Systems Stochastic Local Search (SLS) 4
The 10 commandments for powerful Hybrid Metaheuristics 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. You shall covet your best competitors procedures, methods and strategies, break them into parts and use them as Local Search. 5
Scheme of an Iterative Variation Selection Procedure (IVS) Initialization REPEAT Select Candidate Solutions for Modification Modify Candidate Solutions Select Candidate Solutions for further Iterations UNTIL Stopping Criteria( GNr, LastImprovingGNr, Threshold, ); 6
IVS: Horizontal and Recursive Composition IVS( RecLevel, NS_Set, IVS_ParaSet ) Initialization REPEAT FOR (HLevel = 0) TO GetMaxHLevel( ) DO Select Candidates for Modification Modify Candidates IVS( RecLevel-1, NS_Set, IVS_ParaSet ) Select Candidates for further Iterations UNTIL Stopping Criteria( GNr, LastImprovingGNr, Threshold, Level ); 7
Variation Operators Systematic modification of decision variables Deterministic principals Stochastic principals Local view (i.e. modify only few variables at each step) Global view (i.e. Tree Search) Construction, destruction and modification schemes Decomposition strategies (hierarchical, geographical, functional) Combined or composed variation operators (i.e. VNS, Mutation) 8
Neighborhood Generating Operators Elementary Neighborhood Generating Operator = Systematic parameterized modification of decision variables One Step Neighborhood Neighborhood Transition Graph (NH-Transition Graph) (Asymmetric) Distance measure, metric Neighborhood Search templates Steepest ascent Next ascent K-Step Neighborhood Local optima of quality K (iterative or recursive scheme) Discrepancy Search, Local Branching Rapid-Tree Search, Rapid-B&B Probabilistic K-Step Neighborhood (i.e. Mutation-Operator) 9
Multiple Solution Variation Operators (Recombination) Recombination Operator = Parents define a subspace or a subset of the search space Standard Crossover = Randomly selected point in this subset Series of points in this subset using a NH-Transition Graph Deterministic principles Connecting path between parents (with discrepancies) Enumerate the complete subset Deterministic Sub-Problem Solver Probabilistic principles Re-Sampling or Random Walk Connecting random path (with discrepancies) Probabilistic Sub-Problem Solver 10
Combining IVS Procedures Variable Neighborhood Search Fixed sequence Probabilistic sequence Adaptive or self-adaptive sequence Evolutionary Algorithms Mutation Operator (Probabilistic K-Step Neighborhood) Crossover Operator (Dynamic Sub Problem Search) Hybrid Evolutionary Algorithms i.e. Hybrid (1+1) EA = Iterative Local Search Multi - Start Metaheuristics Number of runs vs. number of iterations (Multi Start Factor) 11
Aspects of Iterative Variation Selection Procedures Problem specific representation Problem specific variation operators (Variable) Neighborhood Search techniques Accelerated Delta Evaluation of the objective function Efficient data structures Dynamic Adaptive Decomposition strategies (DADs) Biased disruption strategies Adaptive or self-adaptive search control Population Management 12
Transportation Logistics: Sub-Problems 13
Vehicle Routing Problems (VRP) "Traveling Salesman Problem" (TSP) CVRP, VRPTW, VRPBH, PDVRP "Open VRP" (OVRP) "Periodic TSP and VRP" (PVRP, PTSP) "Multiple Depot VRP" (MDVRP) "Periodic MDVRP" (PMDVRP) Time Windows (TW) Split Demands (SD) 14
Modeling Concepts Process and Vehicle Oriented View Resource Consumption Concept, Set of Resources, Status Variables (Composed) Transformation Functions, Finite State Machines Route 1: B1 1 2 3 4 5 6 7 E1 B1 C1 C2 Route 2: B2 8 9 10 11 12 13 14 E2 C6 C7 E1 C3 C4 C5 C10 C11 C12 B2 C8 C9 C13 C14 E2 15
Modeling of the OVRP: Structural Constraints Structural constraints that are kept valid by Neighborhood operators: All demands are satisfied Each customer is visited exactly once All tours start at the depot Open tours Open tour ends are modeled with an auxiliary ending node (virtual ending depot) No resource consumption for virtual ending depots and their incoming edges (Transformation function is the Identity Function) Standard CVRP-Operators can be used 16
Modeling of the OVRP: Resource Constraints Capacity Resource initialization at the (starting) depot FreeSpace := Capacity Transformation At node C(i) : FreeSpace := FreeSpace Demand( C(i) ) Tour length restriction Resource initialization at the (starting) depot RemainingTourLength := MaxTourLength Transformation At edges E(i,j): RemainingTourLength := RemainingTourLength DrivingTime( E(i,j) ) At nodes C(i): RemainingTourLength := RemainingTourLength HandlingTime( C(i) ) 17
Examples: Constraints and Modeling Aspects Capacity limit Tour length limit Time Windows Split Demand, Single Unit VRP Pickup and Delivery Backhauls Heterogeneous fleet Multiple compartments, dynamic compartment sizes, load restrictions Fixed costs Customer dependent costs Asymmetric distance and driving costs Customer specific service times, back on route times Traffic flow factor Flexible starting times 18
Operators / Neighborhoods / Neighborhood Size # customers = n, # routes = m Single Route Operators InsertCustomer, RemoveCustomer: O(1) CheapestInsertCustomer: O(n) 2 OPT: O(n²) Multiple Route Operators Single Customer Operators Move: O(n²) Exchange: O(n²) Combined Move/Exchange: O(n²) Path Operators (Multiple adjacent customers, solution parts) Concatenate Tour Pair: O(m²) Split Tour: O(n) Path Exchange: O(n 4 ) Restricted Path Exchange (one end fixed to be a depot): O(n²) 19
Operator: Exchange Path Route 1: B1 1 2 3 4 5 6 7 E1 Route 2: B2 8 9 10 11 12 13 14 E2 B1 C1 C2 C6 C7 E1 C3 C4 C5 C10 C11 C12 B2 C8 C9 C13 C14 E2 20
Operator: Exchange Path = Exchange 2 SuperCustomers Route 1: B1 1 2 3 4 5 6 7 E1 Route 2: B2 8 9 10 11 12 13 14 E2 B1 C1 C2 C6 C7 E1 SC(3,5) C3 C4 C5 SC(10,12) C10 C11 C12 B2 C8 C9 C13 C14 E2 21
Exchange Path = Concatenate 2 x 3 SuperCustemers Route 1: B1 1 2 3 4 5 6 7 E1 Route 2: B2 8 9 10 11 12 13 14 E2 B1 SC(1,2) C1 C2 SC(6,7) C6 C7 E1 SC(3,5) C3 C4 C5 SC(10,12) C10 C11 C12 B2 SC(8,9) C8 C9 SC(13,14) C13 C14 E2 22
Constraint Handling and Acceleration Techniques Super-Customer Concept for Accelerated Delta Function Evaluations of Path Based Neighborhood Generating Operators Super-Customer Matrix, Fast Super-Customer Lookup Object or Hash Tables Reusing information of already visited Neighborhoods and Sub-Neighborhoods Priority Lists Static or Dynamic Neighborhood Reduction, Candidate or Tabu Lists Efficient Data Structures 23
Partial Fixing of Decision Variables Neighborhood Specific Local Optima Flags for parts of the solution: Customers (or subsets of customers) Routes (or subsets of routes) Routes assigned to a depot (or a subset of depots) Routes assigned to a day (or a sub period) Partial solutions according to a decomposition scheme 24
Decomposition in Sub Problems and Large Neighborhoods Hierarchical decomposition PMDVRP, PVRP, PTSP MDVRP, MDTSP CVRP TSP Select a series of different subsets of Routes Geographical decomposition Disjoint (Parallelization) Overlapping VNS-Scheme: Increasing number of routes 25
Scheme of a Hybrid (1+1)-Evolutionary Strategy GNr := 0; LastImprovingGeneration:= 0; Initialization( Parent ); VariableNeighborhoodSearch( Parent ); REPEAT GNr := GNr+1; Child := Mutation( Parent ); VariableNeighborhoodSearch( Child ); if ( Fitness( Child ) => Fitness( Parent ) ) Parent := Child; LastImprovingGeneration:= GNr; UNTIL StoppingCriteria( GNr, LastImprovingGeneration, FitnessThreshhold ); 26
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OVRP Benchmarks 14 CVRP instances of Christofides, Mingozzi and Toth Best known results from Rokpe et al. (2007), Fleszar et al. (2008), Li et al. (2007) and Derigs et al. (2008) instance max. max. tour handling best known results Our results name # customer capacity length time tour count tour length tour count tour length diff c1 50 160-0 6 412,95 6 412,95 0,00% c2 75 140-0 11 564,06 11 564,06 0,00% c3 100 200-0 9 639,25 9 639,25 0,00% c4 150 200-0 12 733,13 12 733,13 0,00% c5 200 200-0 17 869,24 17 868,44-0,10% c6 50 160 200 10 6 412,95 6 412,95 0,00% c7 75 140 160 10 11 568,49 11 566,93-0,28% c8 100 200 230 10 9 644,63 15 640,89-0,59% c9 150 200 200 10 13 757,84 13 741,44-2,17% c10 200 200 200 10 17 875,67 17 871,58-0,47% c11 120 200-0 10 678,54 10 678,54 0,00% c12 100 200-0 10 534,24 10 534,24 0,00% c13 120 200 720 50 11 869,50 11 836,55-3,79% c14 100 200 1040 90 11 591,87 11 552,64-6,63% 32
CVRP vs. OVRP: C8 with tour length limit (100 customers) 33
Are highly optimized VRPTW solutions robust enough? 34
Uncertainty: Fitting Speed with a Cauchy Distribution 1 9 17 25 33 41 49 57 65 73 81 89 97 105 113 121 129 137 145 35
Modeling Uncertainty and Risk with Resources Probabilistic Resource Consumption (i.e. speed, driving time, demands, ) Series of Conditional Probability Distributions C1 C2 C3 C4 C17 C8 C7 C6 C5 Depot C10 C11 C9 C12 C13 C14 C15 C16 36
Risk Assignment and Visualization 37
Tradeoff between Costs and Risk RC 105 0,6 0,5 0,4 Risk 0,3 0,2 SMS-EMOA VRP 3-EMOA 0,1 0 1000 2000 3000 4000 5000 6000 7000 Costs 38
Complexity Size of the Search Space: Exponential Algorithmic Complexity: NP-Hard Complexity Management and Handling: Local Modification, Neighborhood Generating Operators, Local Search Decomposition of Objective Functions in Sub-Functions Neighborhoods into Sub-Neighborhoods Problems into Sub-Problems Re-use of already done partial work (Common) Sub-Functions Fast Delta Function Evaluation (according to a Neighborhood) Overlapping Neighborhoods (Common) Sub-Neighborhoods (Common) Sub-Problems 39
Parallelization Levels Decomposition Level Sub-Functions Sub-Neighborhoods Sub-Problems Sub-Networks Multiple Strategy Level Multiple Parameter Sets Multiple Neighborhoods Multiple Metaheuristics Multi Agent Systems (MAS) 40
Summary and Challenges Abstract and generalized view on Metaheuristics Flexible modeling and optimization concept for Rich VRP Acceleration techniques for path based Neighborhood Search methods A simple way how to quantify risk and to use it in a bi-criteria search method Thank you for your attention! Noise, incomplete and uncertain data, robustness, risk management Dynamic and Online Optimization, Multi-criteria Optimization Enhanced variation mechanisms, adaptive disruption strategies Parallel methods, adaptive strategy control mechanisms (Species, Agents) More complex and hierarchical nested optimization problems 41